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Jumps of Milnor Numbers of Brieskorn–Pham Singularities in Non-Degenerate Families

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Jumps of Milnor Numbers of Brieskorn–Pham Singularities in Non-Degenerate Families Results Math (2018) 73:94 c 2018 The Author(s) 1422-6383/18/030001-13 published online June 18, 2018 Results in Mathematics https://doi.org/10.1007/s00025-018-0849-y Jumps of Milnor Numbers of Brieskorn–Pham Singularities in Non-degenerate Families Tadeusz Krasi´nski and Justyna Walewska Abstract. The jump of the Milnor number of an isolated singularity f0 is the minimal non-zero difference between the Milnor numbers of f0 and one of its deformation (fs). In the case fs are non-degenerate singularities we call the jump non-degenerate. We give a formula (an inductive algorithm using diophantine equations) for the non-degenerate jump of f0 in the case f0 is a convenient singularity with only one (n − 1)-dimensional face of its Newton diagram which equivalently (in our problem) can be replaced by the Brieskorn–Pham singularities. Mathematics Subject Classification. 14B07, 32S30. Keywords. Milnor number, deformation of singularity, non-degenerate singularity, Newton polyhedron, Newton diagram. 1. Introduction n Let f0 :(C , 0) → (C, 0) be an (isolated) singularity, i.e. f0 is the germ at 0 of a holomorphic function having an isolated critical point at 0 ∈ Cn,and 0 ∈ C as the corresponding critical value. More specifically, there exists a ˆ representative f0 : U → C of f0 holomorphic in an open neighborhood U of the point 0 ∈ Cn such that: ˆ • f0(0) = 0, ˆ •∇f0(0) = 0, ˆ •∇f0(z) =0for z ∈ U\{0}, 94 Page 2 of 13 T. Krasi´nski and J. Walewska Results Math where for a holomorphic function f we put ∇f := (∂f/∂z1,...,∂f/∂zn). The ring of germs of holomorphic functions of n variables will be denoted by On. A deformation of the singularity f0 is the germ of a holomorphic function f = f(s, z):(C × Cn, 0) → (C, 0) such that: • f(0,z)=f0(z), • f(s, 0) = 0. The deformation f(s, z) of the singularity f0 will also be treated as a family (fs) of germs, putting fs(z):=f(s, z). Since f0 is an isolated singularity, fs has also isolated singularities near the origin, for sufficiently small s [8, Theorem 2.6 in Chapter I]. Remark 1. Notice that in the deformation (fs) there can occur in particular smooth germs, that is germs satisfying ∇fs(0) = 0. In this context, the symbol ∇fs will always denote ∇zfs(z). By the above assumptions, for every sufficiently small s, one can define a (finite) number μs as the Milnor number of fs, namely μs := μ(fs) = dimC On/(∇fs), where (∇fs) is the ideal in On generated by ∂fs/∂z1,...,∂fs/∂zn. Since the Milnor number is upper semi-continuous in the Zariski topology [8, Proposition 2.57 in Chapter II], there exists an open neighborhood S of the point 0 ∈ C such that • μs =const. for s ∈ S\{0}, • μ0 μs for s ∈ S. The (constant) difference μ0 − μs for s ∈ S\{0} will be called the jump of the deformation (fs) and denoted by λ((fs)). The smallest non-zero value among the jumps of all deformations of the singularity f0 will be called the jump (of the Milnor number) of the singularity f0 and denoted by λ(f0). The first general result concerning the jump was Guse˘ın-Zade’s [9], who proved that there exist singularities f0 for which λ(f0) > 1 and that for irre- ducible plane curve singularities it holds λ(f0)=1.In[4] the authors proved that λ(f0) is not a topological invariant of f0 but it is an invariant of the stable equivalence of singularities. The computation of λ(f0) is not an easy task. It is related to the problem of adjacency of classes of singularities. Only for few classes of singularities we know the exact value of λ(f0). For plane curve sin- gularities (n =2)intheX9 class (see [1] for terminology) that is singularities f a x, y x4 y4 ax2y2 a ∈ C a2 λ f a of the form 0 ( ):= + + , , =4,wehave ( 0 )=2(see f d x, y xd yd d [4]) and for specific homogeneous singularities 0 ( )= + , 2, we λ f d d/ x x have ( 0 )=[ 2], where [ ] is the integer part of (see [5]). nd In the paper we consider a weaker problem: compute the jump λ (f0) of f0 over all non-degenerate deformations of f0 (i.e. the fs in deformations (fs)of nd f0 are non-degenerate singularities). Clearly, we always have λ(f0) λ (f0). Vol. 73 (2018) Jumps of Milnor Numbers of Brieskorn–Pham Singularities Page 3 of 13 94 Up to now, this problem has been studied only for plane curve singularities by Bodin [3], Brzostowski et al. [5] and Walewska [16,17]. We give a formula nd (more precisely an algorithm) in n-dimensional case for λ (f0), where f0 is non-degenerate, convenient and has its Newton diagram reduced to one (n−1)- dimensional face. Since for non-degenerate singularities fs of a deformation (fs)off0 the Milnor number μ(fs) depends only on the Newton diagram of fs and not on specific coefficients of fs (see Sect. 2) we may restrict considerations to the class of the Brieskorn–Pham singularities p f z 1 ··· zpn ,p ,i ,...,n, 0(z)= 1 + + n i 2 =1 (1) where z =(z1,...,zn). Remark 2. Let us note that we compute non-degenerate jump of a singularity in a fixed system of coordinates. The reason is that non-degeneracy depends on 3 3 nd coordinate system. For instance, for singularity f0(x, y)=x + y , λ (f0)=2 in the original system of coordinates while for L : x = x − y, y = y,wehave nd λ (f0 ◦ L)=1. The problem of finding the minimal non-degenerate jump of a singularity in arbitrary system of coordinates is open. 2. Non-degenerate Singularities f In this Section we recall the notion of non-degenerate singularities. Let 0(z)= i n n a z be a singularity and supp(f )={i ∈ N : a =0 } the support of i∈N0 i 0 0 i f f ⊂ Rn f 0.TheNewton polyhedron Γ+( 0) of 0 is the convex hull of the set Rn . i∈supp(f0)(i + +) The finite set of compact faces (of all dimensions) of the boundary of Γ+(f0) constitutes the Newton diagram of f0 and is denoted by Γ(f0). The singularity f0 is convenient if Γ+(f0) intersects all coordinate axes Rn S ∈ f in . For each face Γ( 0) we define a weighted homogeneous polynomial f a i. f S ∈ f ( 0)S(z)= i∈S iz We call the singularity 0 non-degenerate on Γ( 0) if the system of equations ∂(f0)S/∂zi(z)=0,i =1,...,n has no solutions in ∗ n (C ) . The singularity f0 is non-degenerate if f0 is non-degenerate on every face S ∈ Γ(f0). In the sequel we fix the following notations for coordinate hyperplanes in n R .LetI = {i1,...,ik}⊂{1, 2,...,n}. We denote Ox ...x {x x ,...,x ∈ Rn x ,j∈ I} i1 ik := =( 1 n) : j =0 and {x ... x } {x x ,...,x ∈ Rn x ... x }. i1 = = ik =0 := =( 1 n) : i1 = = ik =0 For a convenient singularity f0 we define the Newton number ν(f0)off0 by n n n ν(f0):=n!V − (n − 1)!Vi + (n − 2)!Vij −···+(−1) V12...n, i=1 i,j=1 i<j 94 Page 4 of 13 T. Krasi´nski and J. Walewska Results Math where V is the n-dimensional volume under Γ+(f0), Vi is the (n−1)-dimensional volume under Γ+(f0) on the hyperplane {xi =0}, Vij is the (n−2)-dimensional volume under Γ+(f0) on the hyperplane {xi = xj =0} and so on. The importance of ν(f0) has its source in the celebrated Kouchnirenko theorem. Theorem 2.1 [11]. If f0 is a convenient singularity, then (1) μ(f0) ν(f0), (2) if f0 is non-degenerate then μ(f0)=ν(f0). Remark 3. For non-convenient singularities the Kouchnirenko Theorem also holds provided ν(f0) is appropriately defined (see for instance [6]). It is in- teresting that both conditions of item 2. of the Theorem are equivalent in 2-dimensional case by Ploski [15]. For weaker non-degeneracy conditions equiv- alent to the equality μ(f0)=ν(f0)inn-dimensional case see recent preprint by Mondal [12]. 3. Non-degenerate Jump of Milnor Numbers of Singularities Let f0 ∈On be a singularity. A deformation (fs)off0 is called non-degenerate if fs is non-degenerate for s = 0. The set of all non-degenerate deformations nd nd of the singularity f0 will be denoted by D (f0). Non-degenerate jump λ (f0) of the singularity f0 is the minimal of non-zero jumps over all non-degenerate deformations of f0, which means nd λ (f ) := min λ((fs)), 0 nd (fs)∈D0 (f0) Dnd f f f where by 0 ( 0) we denote all non-degenerate deformations ( s)of 0 for which λ((fs)) =0. Obviously Proposition 3.1. For each singularity f0 we have the inequality nd λ(f0) ≤ λ (f0). From the Kouchnirenko Theorem we get ˜ Corollary 3.2. If f0 and f0 are non-degenerate and convenient singularities ˜ nd nd ˜ and Γ(f0)=Γ(f0) then λ (f0)=λ (f0). Corollary 3.3. If f0 is a convenient and non-degenerate singularity such that the Newton diagram Γ(f0) has only one (n − 1)-dimensional face which inter- sects the axes Ox1,...,Oxn in points (p1, 0,...,0),...,(0,...,0,pn) respec- p f¯ z z 1 ··· zpn tively, then for the Brieskorn–Pham singularity ( )= 1 + + n we have nd nd ¯ λ (f0)=λ (f).
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